Anti-commutative Groebner-Shirshov basis of a free Lie algebra
L. A. Bokut, Yuqun Chen, Yu Li
TL;DR
This paper introduces a new method for constructing a basis of a free Lie algebra using the Composition-Diamond lemma for anti-commutative algebras, offering an alternative to traditional proofs involving Hall words.
Contribution
It presents an anti-commutative Groebner-Shirshov basis approach for free Lie algebras, providing a novel proof technique based on the Composition-Diamond lemma.
Findings
Established an anti-commutative Groebner-Shirshov basis for free Lie algebras
Provided an alternative proof for the basis of free Lie algebras using the Composition-Diamond lemma
Connected classical Hall basis construction with modern algebraic methods
Abstract
One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).
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